It follows from Appendix A that the third component \(I_{3}\)  of the isospin is
conserved in the case of free fermions.
The fermion scattering on the soliton can therefore be regarded as a transition
from the free “in" state, with spatial momentum \(\mathbf {p} = (p, 0)\)  and
isospin \(I_{3}\) , to the free “out" state, with spatial momentum \(\mathbf {p}^{\prime }= (p\cos (\theta ), p\sin (\theta ))\)  and isospin \(I_{3}^{\prime }\) .
A transition without (with) a change in the isospin can be considered to be
elastic (inelastic).
According to the theory of scattering [1]}, [2]}, the scattering
state that corresponds to the “in" isospin \(I_{3}\)  and the “out" isospin
\(I_{3}^{\prime }\)  is described asymptotically by the wave function
\(\Psi _{I_{3}^{\prime },I_{3}} \sim \psi _{\varepsilon ,\mathbf {p},I_{3}}\delta _{I_{3}^{\prime },I_{3}}+\frac{1}{\sqrt{2\varepsilon }}u_{\varepsilon ,\mathbf {p}^{\prime },I_{3}^{\prime }\,}f_{I_{3}^{\prime },I_{3}}\left(p, \theta \right)\frac{e^{ipr}}{\sqrt{-ir}}, \) 
